An Improved Parallel Iterative Algorithm for Stable Matching
Colin White, Undergraduate
Dr. Enyue Lu, Faculty Mentor
Depts. of Math & Computer Science
Amherst College
Amherst, USA
[email protected]
Dept. of Math. & Computer Science
Salisbury University
Salisbury, USA
[email protected]
Abstract— We propose a parallel algorithm for finding a stable
matching that converges in O(n log n) average time using n2
processors. The algorithm is based on the Parallel Iterative
Improvement (PII) algorithm, which finds a stable matching
with approximately a 90% success rate. Our algorithm, called
the PII-SC algorithm, uses a smart initiation method that
decreases the number of iterations to find a stable matching,
and also applies a cycle detection method to find a stable
matching based on patterns in the preference lists. Both
methods decrease the number of times it fails to find a stable
matching by three orders of magnitude, and when combined,
the chance of failure is less than 10-7.
Keywords—Stable Matching, Stable Marriage, Graph
Algorithms, Parallel Algorithms
I.
INTRODUCTION
The stable matching (or stable marriage) problem was
first introduced by Gale and Shapley [1]. Given n men, n
women, and 2n preference lists in which each person ranks
all members of the opposite sex by preference, a matching is
an unordered set of n pairs of a man and woman in which
each person is in exactly one pair. A matching is unstable if
there exists a man and a woman who are not paired with
each other, but both prefer each other to their current
partner. Otherwise, a matching is stable. Gale and Shapley
presented an O(n2) algorithm to find a stable matching from
any preference lists, thus proving that a stable matching
must always exist. The stable matching problem has a wide
variety of applications, from assigning doctors to hospitals,
to real-time switch scheduling, to problems in economics
[2,6].
II.
PII ALGORITHM
The original PII algorithm was designed for timesensitive switch scheduling network problems [4]. It
consists of two phases: the initiation phase and the iteration
phase. It starts by generating a random initial matching in
the initiation phase, and then iteratively picks the “most”
unstable pairs, collectively called the set NM1 of type-1 new
matching pairs (nm1-pairs), to replace current matching
pairs in the iteration phase. Since each nm1-pair contains a
man and a woman from two different current matching
This work is funded by NSF CCF-1156509 under Research Experiences
for Undergraduates Program. C. White did his work as a REU (Research
Experiences for Undergraduates) student at Salisbury University during
the summer of 2013.
pairs, often other pairs must be added until we have n pairs
again. Such pairs, which make up the set NM2 of type-2
new matching pairs (nm2-pairs), are sometimes difficult to
find because of long chains of nm1-pairs that overlap with
respect to the replaced pairs. The nm2-pairs, which may not
be favorable, can cause new unstable pairs in the new
matching, which in turn allows the possibility of cycling in
the iteration phase. The PII algorithm has a 90% chance of
finding a stable matching within n iterations (the vast
majority of failures are due to cycling), and runs in O(n log
n) average time with n2 processors [3,4].
III.
NEW METHODS
In this paper, we give two major improvements that
increase the performance of the PII algorithm: smart
initiation and cycle detection.
A. Smart Initiation
The first major improvement is in the initiation phase:
we use a simple method to find an initial matching that
performs much better than a random matching. We
accomplish this in a way similar to the Gale-Shapley
algorithm: pair each man with their top-ranked woman,
breaking ties with the woman’s preference. Then each
unpaired man gets paired with their next best choice,
however, unlike in the Gale-Shapley algorithm, they cannot
propose to women that are currently already paired with
someone. We continue in this way until we have a
matching.
We claim that the smart initiation method can be done in
linear time with n2 processors as follows. There are at most
n steps before every person is provably paired up. During
each step, there are at most n proposals from the men.
Every woman can pick her most preferable proposal in
constant time using constant-time find-minimum algorithm,
which finds the minimum of k elements using k2 processors
[5]. This causes every step to take constant time, making the
whole method run in linear time.
The smart initiation method decreases the chance of the
algorithm failing to find a stable matching from 1 in 10 to 1
in 20. Furthermore, when the method is run in parallel with
the woman-optimal version of the same method (in which
the roles of men and women are switched), the chances of
failing are reduced to roughly 1 in 400.
B. Cycle Detection
The second major improvement is in the iteration phase:
an additional step that detects if the algorithm is cycling and
then outputs the stable matching. When the algorithm has
reached 3n iterations without finding a stable matching,
experimentation has shown that the matchings generated by
the iteration phase must be cycling. It then saves the current
matching and checks the matching generated in each
successive iteration to see if it is the same as the one in the
3nth iteration. Meanwhile, the algorithm starts creating
chains of nm1-pairs: each nm1-pair either starts a new chain,
or is added to an existing chain if it shares the same man or
woman with the last pair that was added. After the 3nth
matching is detected again, we check if any nm1 chains can
merge together into nm1-cycles. We then pick every other
nm1 pair in the cycles, which results in a stable matching
roughly 9,999 times out of 10,000. The cycle detection
phase can be done in constant time, which does not change
the runtime of the full algorithm.
IV.
N
GaleShapley
Original
PII
Smart
Initiation
Cycle
Detection
PII-SC
Algorithm
10
0.9609
0.9809
0.9999
1.0000
1.0000000
20
0.8953
0.9513
0.9993
1.0000
1.0000000
30
0.8566
0.9226
0.9996
1.0000
0.9999998
40
0.8338
0.9007
0.9988
0.9999
0.9999996
50
0.8198
0.8801
0.9981
0.9999
0.9999994
60
0.8086
0.8719
0.9979
0.9998
0.9999994
70
0.8007
0.8650
0.9986
0.9998
0.9999992
80
0.7935
0.8553
0.9984
0.9998
0.9999989
90
0.7903
0.8642
0.9980
0.9997
0.9999992
100
0.7821
0.8579
0.9980
0.9998
0.9999985
Table 2: Number of successes for finding a stable matching
with n=100 for various iterations per 100,000 trials
RESULTS AND DISCUSSION
We have implemented the PII algorithm with the two
major improvements (which we call the PII-SC algorithm),
along with the PII algorithm with smart initiation, the PII
algorithm with cycle detection, the original PII algorithm,
and the Gale-Shapley algorithm for comparison. For each
trial, we generate random preference lists and run each of the
different algorithms, outputting the success rate and the
number of iterations taken to reach a stable matching. In
table 1, we calculate the success rate for 5n iterations,
varying n from 10 to 100. We ran the PII-SC algorithm for
10 million trials in order to calculate an accurate probability,
but to save computing time, we deemed 100,000 trials each
was sufficient for the rest of the algorithms. Each of the two
major improvements decreases the probability of failure by
multiple orders of magnitude. In table 2, we calculate the
number of successes with n=100, varying the number of
iterations from 0.5n to 5n. The cycle detection algorithm and
PII-SC algorithm only start looking for cycles after 3n
iterations, so we see a major increase in the rate of success at
that time.
V.
Table 1: Probability of finding a stable matching in 5n
iterations with various n values
ONGOING WORK
We are working on classifying cases where the cycle
detection algorithm sometimes fails to find a stable
matching. We have identified different categories of “bad”
cases and why each of them fails to output a stable
matching. If we can generalize the cycle detection algorithm
to account for these cases, along with proving that an initial
matching must either become stable or cycle after 3n
iterations, and that the longest cycle length is 2n, then we
would have an O(n log n) algorithm for stable matching
with n2 processors. We are also working towards finding a
similar algorithm that uses n3 processors and runs in linear
time.
Iterations
GaleShapley
Original
PII
Smart
Initiation
Cycle
Detection
PII-SC
Algorithm
0.5n
93
81846
86166
81846
86166
n
5308
86363
95269
86363
95269
1.5n
19392
86425
99347
86425
99347
2n
35522
86427
99777
86427
99777
2.5n
50128
86427
99784
86427
99784
3n
61998
86427
99784
86427
99784
3.5n
71012
86427
99784
99981
100000
4n
78063
86427
99784
99982
100000
4.5n
83483
86427
99784
99982
100000
5n
87609
86427
99784
99982
100000
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Gale, D. and Shapley, L.S.: “College admissions and the stability of
marriage”, American Mathematical Monthly, Vol. 69 (1962) 9–15
Gusfield, D. and Irving, R.W.: The Stable Marriage Problem
Structure and Algorithms, MIT Press (1989)
Korakakis, E.: Examining the Parallelization Limits of the Stable
Matching Problem, Master’s Thesis, University of Edinburgh
(2005)
Lu E. and Zheng S. Q.: "A Parallel Iterative Improvement Stable
Matching Algorithm", High performance computing, Lecture Notes
in Computer Science, Springer-Verlag, Vol. 2913 (2003) 55-65
Quinn, Michael J.: Parallel Computing: Theory and Practice, 2nd
ed., New York: McGraw-Hill, 1994
The Prize in Economic Sciences 2012, “Stable matching: Theory,
evidence, and practical design”, http://www.nobelprize.org
/nobel_prizes/economic-sciences/laureates/2012/populareconomicsciences2012.pdf